Deep Learning of Turbulent Scalar Mixing

Based on recent developments in physics-informed deep learning and deep hidden physics models, we put forth a framework for discovering turbulence models from scattered and potentially noisy spatio-temporal measurements of the probability density function (PDF). The models are for the conditional expected diffusion and the conditional expected dissipation of a Fickian scalar described by its transported single-point PDF equation. The discovered model are appraised against exact solution derived by the amplitude mapping closure (AMC)/ Johnsohn-Edgeworth translation (JET) model of binary scalar mixing in homogeneous turbulence.Comment: arXiv admin note: text overlap with arXiv:1808.04327, arXiv:1808.0895

Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit

In deep latent Gaussian models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. This work considers the diffusion limit of such models, where the number of layers tends to infinity, while the step size and the noise variance tend to zero. The limiting latent object is an It\^o diffusion process that solves a stochastic differential equation (SDE) whose drift and diffusion coefficient are implemented by neural nets. We develop a variational inference framework for these \textit{neural SDEs} via stochastic automatic differentiation in Wiener space, where the variational approximations to the posterior are obtained by Girsanov (mean-shift) transformation of the standard Wiener process and the computation of gradients is based on the theory of stochastic flows. This permits the use of black-box SDE solvers and automatic differentiation for end-to-end inference. Experimental results with synthetic data are provided

Monge-Amp\`ere Flow for Generative Modeling

We present a deep generative model, named Monge-Amp\`ere flow, which builds on continuous-time gradient flow arising from the Monge-Amp\`ere equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Amp\`ere flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative model by designing suitable scalar potential functions. We apply the approach to unsupervised density estimation of the MNIST dataset and variational calculation of the two-dimensional Ising model at the critical point. This approach brings insights and techniques from Monge-Amp\`ere equation, optimal transport, and fluid dynamics into reversible flow-based generative models

Deep Latent-Variable Kernel Learning

Deep kernel learning (DKL) leverages the connection between Gaussian process (GP) and neural networks (NN) to build an end-to-end, hybrid model. It combines the capability of NN to learn rich representations under massive data and the non-parametric property of GP to achieve automatic regularization that incorporates a trade-off between model fit and model complexity. However, the deterministic encoder may weaken the model regularization of the following GP part, especially on small datasets, due to the free latent representation. We therefore present a complete deep latent-variable kernel learning (DLVKL) model wherein the latent variables perform stochastic encoding for regularized representation. We further enhance the DLVKL from two aspects: (i) the expressive variational posterior through neural stochastic differential equation (NSDE) to improve the approximation quality, and (ii) the hybrid prior taking knowledge from both the SDE prior and the posterior to arrive at a flexible trade-off. Intensive experiments imply that the DLVKL-NSDE performs similarly to the well calibrated GP on small datasets, and outperforms existing deep GPs on large datasets.Comment: 13 pages, 8 figures, preprint under revie

Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data

We present hidden fluid mechanics (HFM), a physics informed deep learning framework capable of encoding an important class of physical laws governing fluid motions, namely the Navier-Stokes equations. In particular, we seek to leverage the underlying conservation laws (i.e., for mass, momentum, and energy) to infer hidden quantities of interest such as velocity and pressure fields merely from spatio-temporal visualizations of a passive scaler (e.g., dye or smoke), transported in arbitrarily complex domains (e.g., in human arteries or brain aneurysms). Our approach towards solving the aforementioned data assimilation problem is unique as we design an algorithm that is agnostic to the geometry or the initial and boundary conditions. This makes HFM highly flexible in choosing the spatio-temporal domain of interest for data acquisition as well as subsequent training and predictions. Consequently, the predictions made by HFM are among those cases where a pure machine learning strategy or a mere scientific computing approach simply cannot reproduce. The proposed algorithm achieves accurate predictions of the pressure and velocity fields in both two and three dimensional flows for several benchmark problems motivated by real-world applications. Our results demonstrate that this relatively simple methodology can be used in physical and biomedical problems to extract valuable quantitative information (e.g., lift and drag forces or wall shear stresses in arteries) for which direct measurements may not be possible

ODE2^2VAE: Deep generative second order ODEs with Bayesian neural networks

We present Ordinary Differential Equation Variational Auto-Encoder (ODE$^2$VAE), a latent second order ODE model for high-dimensional sequential data. Leveraging the advances in deep generative models, ODE$^2$VAE can simultaneously learn the embedding of high dimensional trajectories and infer arbitrarily complex continuous-time latent dynamics. Our model explicitly decomposes the latent space into momentum and position components and solves a second order ODE system, which is in contrast to recurrent neural network (RNN) based time series models and recently proposed black-box ODE techniques. In order to account for uncertainty, we propose probabilistic latent ODE dynamics parameterized by deep Bayesian neural networks. We demonstrate our approach on motion capture, image rotation and bouncing balls datasets. We achieve state-of-the-art performance in long term motion prediction and imputation tasks

Deep Learning of Vortex Induced Vibrations

Vortex induced vibrations of bluff bodies occur when the vortex shedding frequency is close to the natural frequency of the structure. Of interest is the prediction of the lift and drag forces on the structure given some limited and scattered information on the velocity field. This is an inverse problem that is not straightforward to solve using standard computational fluid dynamics (CFD) methods, especially since no information is provided for the pressure. An even greater challenge is to infer the lift and drag forces given some dye or smoke visualizations of the flow field. Here we employ deep neural networks that are extended to encode the incompressible Navier-Stokes equations coupled with the structure's dynamic motion equation. In the first case, given scattered data in space-time on the velocity field and the structure's motion, we use four coupled deep neural networks to infer very accurately the structural parameters, the entire time-dependent pressure field (with no prior training data), and reconstruct the velocity vector field and the structure's dynamic motion. In the second case, given scattered data in space-time on a concentration field only, we use five coupled deep neural networks to infer very accurately the vector velocity field and all other quantities of interest as before. This new paradigm of inference in fluid mechanics for coupled multi-physics problems enables velocity and pressure quantification from flow snapshots in small subdomains and can be exploited for flow control applications and also for system identification.Comment: arXiv admin note: text overlap with arXiv:1808.0432

Data recovery in computational fluid dynamics through deep image priors

One of the challenges encountered by computational simulations at exascale is the reliability of simulations in the face of hardware and software faults. These faults, expected to increase with the complexity of the computational systems, will lead to the loss of simulation data and simulation failure and are currently addressed through a checkpoint-restart paradigm. Focusing specifically on computational fluid dynamics simulations, this work proposes a method that uses a deep convolutional neural network to recover simulation data. This data recovery method (i) is agnostic to the flow configuration and geometry, (ii) does not require extensive training data, and (iii) is accurate for very different physical flows. Results indicate that the use of deep image priors for data recovery is more accurate than standard recovery techniques, such as the Gaussian process regression, also known as Kriging. Data recovery is performed for two canonical fluid flows: laminar flow around a cylinder and homogeneous isotropic turbulence. For data recovery of the laminar flow around a cylinder, results indicate similar performance between the proposed method and Gaussian process regression across a wide range of mask sizes. For homogeneous isotropic turbulence, data recovery through the deep convolutional neural network exhibits an error in relevant turbulent quantities approximately three times smaller than that for the Gaussian process regression,. Forward simulations using recovered data illustrate that the enstrophy decay is captured within 10% using the deep convolutional neural network approach. Although demonstrated specifically for data recovery of fluid flows, this technique can be used in a wide range of applications, including particle image velocimetry, visualization, and computational simulations of physical processes beyond the Navier-Stokes equations

On Wasserstein Reinforcement Learning and the Fokker-Planck equation

Policy gradients methods often achieve better performance when the change in policy is limited to a small Kullback-Leibler divergence. We derive policy gradients where the change in policy is limited to a small Wasserstein distance (or trust region). This is done in the discrete and continuous multi-armed bandit settings with entropy regularisation. We show that in the small steps limit with respect to the Wasserstein distance $W_2$, policy dynamics are governed by the Fokker-Planck (heat) equation, following the Jordan-Kinderlehrer-Otto result. This means that policies undergo diffusion and advection, concentrating near actions with high reward. This helps elucidate the nature of convergence in the probability matching setup, and provides justification for empirical practices such as Gaussian policy priors and additive gradient noise

A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems

We propose a new composite neural network (NN) that can be trained based on multi-fidelity data. It is comprised of three NNs, with the first NN trained using the low-fidelity data and coupled to two high-fidelity NNs, one with activation functions and another one without, in order to discover and exploit nonlinear and linear correlations, respectively, between the low-fidelity and the high-fidelity data. We first demonstrate the accuracy of the new multi-fidelity NN for approximating some standard benchmark functions but also a 20-dimensional function. Subsequently, we extend the recently developed physics-informed neural networks (PINNs) to be trained with multi-fidelity data sets (MPINNs). MPINNs contain four fully-connected neural networks, where the first one approximates the low-fidelity data, while the second and third construct the correlation between the low- and high-fidelity data and produce the multi-fidelity approximation, which is then used in the last NN that encodes the partial differential equations (PDEs). Specifically, in the two high-fidelity NNs a relaxation parameter is introduced, which can be optimized to combine the linear and nonlinear sub-networks. By optimizing this parameter, the present model is capable of learning both the linear and complex nonlinear correlations between the low- and high-fidelity data adaptively. By training the MPINNs, we can:(1) obtain the correlation between the low- and high-fidelity data, (2) infer the quantities of interest based on a few scattered data, and (3) identify the unknown parameters in the PDEs. In particular, we employ the MPINNs to learn the hydraulic conductivity field for unsaturated flows as well as the reactive models for reactive transport. The results demonstrate that MPINNs can achieve relatively high accuracy based on a very small set of high-fidelity data